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Fluid dynamicsFrom Wikipedia, the free encyclopedia
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Continuum mechanics
Laws[show]
Solid mechanics [show]
Fluid mechanics [hide]
Fluids
Fluid statics
Fluid dynamics
Navier–Stokes equations
Bernoulli's principle
Poiseuille equation
Buoyancy
Viscosity
Newtonian
Non-Newtonian
Archimedes' principle
Pascal's law
Pressure
Liquids
Surface tension
Capillary action
Gases
Atmosphere
Boyle's law
Charles's law
Gay-Lussac's law
Combined gas law
Plasma
Rheology [show]
Scientists[show]
V
T
E
Typical aerodynamic teardrop shape, assuming a viscousmedium passing from left to right, the diagram shows the
pressure distribution as the thickness of the black line and shows the velocity in the boundary layer as the violet
triangles. The green vortex generators prompt the transition to turbulent flow and prevent back-flow also called flow
separation from the high pressure region in the back. The surface in front is as smooth as possible or even
employsshark like skin, as any turbulence here will reduce the energy of the airflow. The truncation on the right, known
as aKammback, also prevents back flow from the high pressure region in the back across the spoilers to the convergent
part.
In physics, fluid dynamics is a subdiscipline of fluid mechanics that deals with fluid flow—the natural
science of fluids (liquids and gases) in motion. It has several subdisciplines itself,
including aerodynamics (the study of air and other gases in motion) and hydrodynamics (the study of
liquids in motion). Fluid dynamics has a wide range of applications, including
calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines,
predicting weather patterns, understanding nebulae in interstellar space and reportedly modelling fission
weapondetonation. Some of its principles are even used in traffic engineering, where traffic is treated as a
continuous fluid.
Fluid dynamics offers a systematic structure—which underlies these practical disciplines—that embraces
empirical and semi-empirical laws derived from flow measurement and used to solve practical problems.
The solution to a fluid dynamics problem typically involves calculating various properties of the fluid, such
as velocity, pressure, density, and temperature, as functions of space and time.
Before the twentieth century, hydrodynamics was synonymous with fluid dynamics. This is still reflected in
names of some fluid dynamics topics, likemagnetohydrodynamics and hydrodynamic stability, both of
which can also be applied to gases.[1]
Contents
[hide]
1 Equations of fluid dynamics
o 1.1 Conservation laws
o 1.2 Compressible vs incompressible flow
o 1.3 Viscous vs inviscid flow
o 1.4 Steady vs unsteady flow
o 1.5 Laminar vs turbulent flow
o 1.6 Newtonian vs non-Newtonian fluids
o 1.7 Subsonic vs transonic, supersonic and hypersonic flows
o 1.8 Magnetohydrodynamics
o 1.9 Other approximations
2 Terminology in fluid dynamics
o 2.1 Terminology in incompressible fluid dynamics
o 2.2 Terminology in compressible fluid dynamics
3 See also
o 3.1 Fields of study
o 3.2 Mathematical equations and concepts
o 3.3 Types of fluid flow
o 3.4 Fluid properties
o 3.5 Fluid phenomena
o 3.6 Applications
o 3.7 Fluid dynamics journals
o 3.8 Miscellaneous
4 Notes
5 References
6 External links
Equations of fluid dynamics[edit]
The foundational axioms of fluid dynamics are the conservation laws, specifically, conservation of
mass, conservation of linear momentum(also known as Newton's Second Law of Motion), and conservation
of energy (also known as First Law of Thermodynamics). These are based on classical mechanics and are
modified inquantum mechanics and general relativity. They are expressed using the Reynolds Transport
Theorem.
In addition to the above, fluids are assumed to obey the continuum assumption. Fluids are composed of
molecules that collide with one another and solid objects. However, the continuum assumption considers
fluids to be continuous, rather than discrete. Consequently, properties such as density, pressure,
temperature, and velocity are taken to be well-defined at infinitesimallysmall points, and are assumed to
vary continuously from one point to another. The fact that the fluid is made up of discrete molecules is
ignored.
For fluids which are sufficiently dense to be a continuum, do not contain ionized species, and have
velocities small in relation to the speed of light, the momentum equations for Newtonian fluidsare
the Navier –Stokes equations , which is a non-linear set of differential equations that describes the flow of a
fluid whose stress depends linearly on velocity gradients and pressure. The unsimplified equations do not
have a general closed-form solution, so they are primarily of use in Computational Fluid Dynamics. The
equations can be simplified in a number of ways, all of which make them easier to solve. Some of them
allow appropriate fluid dynamics problems to be solved in closed form.
In addition to the mass, momentum, and energy conservation equations, a thermodynamical equation of
state giving the pressure as a function of other thermodynamic variables for the fluid is required to
completely specify the problem. An example of this would be the perfect gas equation of state:
where p is pressure, ρ is density, Ru is the gas constant, M is the molar
mass and T is temperature.
Conservation laws[edit]
Three conservation laws are used to solve fluid dynamics problems,
and may be written in integral or differential form. Mathematical
formulations of these conservation laws may be interpreted by
considering the concept of a control volume. A control volume is a
specified volume in space through which air can flow in and out.
Integral formulations of the conservation laws consider the change in
mass, momentum, or energy within the control volume. Differential
formulations of the conservation laws apply Stokes' theorem to yield an
expression which may be interpreted as the integral form of the law
applied to an infinitesimal volume at a point within the flow.
Mass continuity (conservation of mass): The rate of change of fluid
mass inside a control volume must be equal to the net rate of fluid
flow into the volume. Physically, this statement requires that mass
is neither created nor destroyed in the control volume,[2] and can
be translated into the integral form of the continuity equation:
Above, is the fluid density, u is a velocity vector, and t is time.
The left-hand side of the above expression contains a triple integral
over the control volume, whereas the right-hand side contains a
surface integral over the surface of the control volume. The
differential form of the continuity equation is, by the divergence
theorem:
Conservation of momentum : This equation
applies Newton's second law of motion to the control
volume, requiring that any change in momentum of
the air within a control volume be due to the net flow
of air into the volume and the action of external
forces on the air within the volume. In the integral
formulation of this equation, body forces here are
represented by fbody, the body force per unit
mass. Surface forces, such as viscous forces, are
represented by , the net force due
to stresses on the control volume surface.
The differential form of the momentum conservation equation is as
follows. Here, both surface and body forces are accounted for in
one total force, F. For example, F may be expanded into an
expression for the frictional and gravitational forces acting on an
internal flow.
In aerodynamics, air is assumed to be a Newtonian fluid, which
posits a linear relationship between the shear stress (due to
internal friction forces) and the rate of strain of the fluid. The
equation above is a vector equation: in a three dimensional flow, it
can be expressed as three scalar equations. The conservation of
momentum equations for the compressible, viscous flow case are
called the Navier–Stokes equations.
Conservation of energy :
Although energy can be converted
from one form to another, the
total energy in a given closed system
remains constant.
Above, h is enthalpy, k is the thermal conductivity of the fluid, T is
temperature, and is the viscous dissipation function. The viscous
dissipation function governs the rate at which mechanical energy of
the flow is converted to heat. The second law of
thermodynamics requires that the dissipation term is always
positive: viscosity cannot create energy within the control volume.
[3] The expression on the left side is a material derivative.
Compressible vs incompressible flow[edit]
All fluids are compressible to
some extent, that is, changes in
pressure or temperature will result
in changes in density. However, in
many situations the changes in
pressure and temperature are
sufficiently small that the changes
in density are negligible. In this
case the flow can be modelled as
an incompressible flow. Otherwise
the more general compressible
flow equations must be used.
Mathematically, incompressibility
is expressed by saying that the
density ρ of a fluid parcel does not
change as it moves in the flow
field, i.e.,
where D/Dt is the substantial
derivative, which is the sum
of local and convective
derivatives. This additional
constraint simplifies the
governing equations,
especially in the case when
the fluid has a uniform
density.
For flow of gases, to
determine whether to use
compressible or
incompressible fluid
dynamics, the Mach
number of the flow is to be
evaluated. As a rough guide,
compressible effects can be
ignored at Mach numbers
below approximately 0.3. For
liquids, whether the
incompressible assumption is
valid depends on the fluid
properties (specifically the
critical pressure and
temperature of the fluid) and
the flow conditions (how close
to the critical pressure the
actual flow pressure
becomes). Acoustic problems
always require allowing
compressibility, since sound
waves are compression
waves involving changes in
pressure and density of the
medium through which they
propagate.
Viscous vs inviscid flow[edit]
Potential flow around a wing
Viscous problems are those in
which fluid friction has
significant effects on the fluid
motion.
The Reynolds number, which
is a ratio between inertial and
viscous forces, can be used
to evaluate whether viscous
or inviscid equations are
appropriate to the problem.
Stokes flow is flow at very low
Reynolds numbers, Re<<1,
such that inertial forces can
be neglected compared to
viscous forces.
On the contrary, high
Reynolds numbers indicate
that the inertial forces are
more significant than the
viscous (friction) forces.
Therefore, we may assume
the flow to be an inviscid flow ,
an approximation in which we
neglect viscosity completely,
compared to inertial terms.
This idea can work fairly well
when the Reynolds number is
high. However, certain
problems such as those
involving solid boundaries,
may require that the viscosity
be included. Viscosity often
cannot be neglected near
solid boundaries because
the no-slip condition can
generate a thin region of large
strain rate (known
as Boundary layer) which
enhances the effect of even a
small amount of viscosity, and
thus generating vorticity.
Therefore, to calculate net
forces on bodies (such as
wings) we should use viscous
flow equations. As illustrated
by d'Alembert's paradox , a
body in an inviscid fluid will
experience no drag force. The
standard equations of inviscid
flow are the Euler equations.
Another often used model,
especially in computational
fluid dynamics, is to use the
Euler equations away from
the body and the boundary
layer equations, which
incorporates viscosity, in a
region close to the body.
The Euler equations can be
integrated along a streamline
to get Bernoulli's equation.
When the flow is
everywhere irrotational and
inviscid, Bernoulli's equation
can be used throughout the
flow field. Such flows are
called potential flows.
Steady vs unsteady flow[edit]
Hydrodynamics simulation of
the Rayleigh–Taylor
instability [4]
When all the time derivatives
of a flow field vanish, the flow
is considered to be a steady
flow. Steady-state flow refers
to the condition where the
fluid properties at a point in
the system do not change
over time. Otherwise, flow is
called unsteady (also called
transient[5]). Whether a
particular flow is steady or
unsteady, can depend on the
chosen frame of reference.
For instance, laminar flow
over a sphere is steady in the
frame of reference that is
stationary with respect to the
sphere. In a frame of
reference that is stationary
with respect to a background
flow, the flow is unsteady.
Turbulent flows are unsteady
by definition. A turbulent flow
can, however, be statistically
stationary. According to Pope:
[6]
The random
field U(x,t) is
statistically stationary
if all statistics are
invariant under a shift
in time.
This roughly means that all
statistical properties are
constant in time. Often, the
mean field is the object of
interest, and this is constant
too in a statistically stationary
flow.
Steady flows are often more
tractable than otherwise
similar unsteady flows. The
governing equations of a
steady problem have one
dimension fewer (time) than
the governing equations of
the same problem without
taking advantage of the
steadiness of the flow field.
Laminar vs turbulent flow[edit]
Turbulence is flow
characterized by
recirculation, eddies, and
apparent randomness. Flow
in which turbulence is not
exhibited is called laminar. It
should be noted, however,
that the presence of eddies or
recirculation alone does not
necessarily indicate turbulent
flow—these phenomena may
be present in laminar flow as
well. Mathematically, turbulent
flow is often represented via
a Reynolds decomposition, in
which the flow is broken down
into the sum of
an average component and a
perturbation component.
It is believed that turbulent
flows can be described well
through the use of
the Navier –Stokes
equations. Direct numerical
simulation (DNS), based on
the Navier–Stokes equations,
makes it possible to simulate
turbulent flows at moderate
Reynolds numbers.
Restrictions depend on the
power of the computer used
and the efficiency of the
solution algorithm. The results
of DNS have been found to
agree well with experimental
data for some flows.[7]
Most flows of interest have
Reynolds numbers much too
high for DNS to be a viable
option,[8] given the state of
computational power for the
next few decades. Any flight
vehicle large enough to carry
a human (L > 3 m), moving
faster than 72 km/h (20 m/s)
is well beyond the limit of
DNS simulation (Re = 4
million). Transport aircraft
wings (such as on an Airbus
A300 or Boeing 747) have
Reynolds numbers of 40
million (based on the wing
chord). In order to solve these
real-life flow problems,
turbulence models will be a
necessity for the foreseeable
future. Reynolds-averaged
Navier–Stokes
equations (RANS) combined
with turbulence
modelling provides a model of
the effects of the turbulent
flow. Such a modelling mainly
provides the additional
momentum transfer by
the Reynolds stresses,
although the turbulence also
enhances the heat and mass
transfer. Another promising
methodology is large eddy
simulation (LES), especially in
the guise of detached eddy
simulation (DES)—which is a
combination of RANS
turbulence modelling and
large eddy simulation.
Newtonian vs non-Newtonian fluids[edit]
Sir Isaac Newton showed
how stress and the rate
of strain are very close to
linearly related for many
familiar fluids, such
as water and air.
These Newtonian fluids are
modelled by a coefficient
called viscosity, which
depends on the specific fluid.
However, some of the other
materials, such as emulsions
and slurries and some visco-
elastic materials (e.g. blood,
some polymers), have more
complicated non-Newtonian st
ress-strain behaviours. These
materials include sticky
liquids such as latex, honey,
and lubricants which are
studied in the sub-discipline
of rheology.
Subsonic vs transonic, supersonic and hypersonic flows[edit]
While many terrestrial flows
(e.g. flow of water through a
pipe) occur at low mach
numbers, many flows of
practical interest (e.g. in
aerodynamics) occur at high
fractions of the Mach Number
M=1 or in excess of it
(supersonic flows). New
phenomena occur at these
Mach number regimes (e.g.
shock waves for supersonic
flow, transonic instability in a
regime of flows with M nearly
equal to 1, non-equilibrium
chemical behaviour due to
ionization in hypersonic flows)
and it is necessary to treat
each of these flow regimes
separately.
Magnetohydrodynamics[edit]
Main
article: Magnetohydrodynamic
s
Magnetohydrodynamics is the
multi-disciplinary study of the
flow of electrically
conducting fluids
in electromagnetic fields.
Examples of such fluids
include plasmas, liquid
metals, and salt water. The
fluid flow equations are solved
simultaneously with Maxwell's
equations of
electromagnetism.
Other approximations[edit]
There are a large number of
other possible approximations
to fluid dynamic problems.
Some of the more commonly
used are listed below.
The Boussinesq
approximation neglects
variations in density
except to
calculate buoyancy force
s. It is often used in
free convection problems
where density changes
are small.
Lubrication
theory and Hele –Shaw
flow exploits the
large aspect ratio of the
domain to show that
certain terms in the
equations are small and
so can be neglected.
Slender-body theory is
a methodology used
in Stokes flow problems
to estimate the force on,
or flow field around, a
long slender object in a
viscous fluid.
The shallow-water
equations can be used
to describe a layer of
relatively inviscid fluid
with a free surface, in
which
surface gradients are
small.
The Boussinesq
equations are applicable
to surface waves on
thicker layers of fluid and
with steeper
surface slopes.
Darcy's law is used for
flow in porous media,
and works with variables
averaged over several
pore-widths.
In rotating systems,
the quasi-geostrophic
approximation assumes
an almost perfect
balance
between pressure
gradients and the Coriolis
force. It is useful in the
study of atmospheric
dynamics.
Terminology in fluid dynamics[edit]
The concept of pressure is
central to the study of both
fluid statics and fluid
dynamics. A pressure can be
identified for every point in a
body of fluid, regardless of
whether the fluid is in motion
or not. Pressure can
be measured using an
aneroid, Bourdon tube,
mercury column, or various
other methods.
Some of the terminology that
is necessary in the study of
fluid dynamics is not found in
other similar areas of study. In
particular, some of the
terminology used in fluid
dynamics is not used influid
statics.
Terminology in incompressible fluid dynamics[edit]
The concepts of total
pressure and dynamic
pressure arise
from Bernoulli's equation and
are significant in the study of
all fluid flows. (These two
pressures are not pressures
in the usual sense—they
cannot be measured using an
aneroid, Bourdon tube or
mercury column.) To avoid
potential ambiguity when
referring to pressure in fluid
dynamics, many authors use
the term static pressure to
distinguish it from total
pressure and dynamic
pressure. Static pressure is
identical to pressure and can
be identified for every point in
a fluid flow field.
In Aerodynamics, L.J. Clancy
writes:[9] To distinguish it from
the total and dynamic
pressures, the actual
pressure of the fluid, which is
associated not with its motion
but with its state, is often
referred to as the static
pressure, but where the term
pressure alone is used it
refers to this static pressure.
A point in a fluid flow where
the flow has come to rest (i.e.
speed is equal to zero
adjacent to some solid body
immersed in the fluid flow) is
of special significance. It is of
such importance that it is
given a special name—
a stagnation point. The static
pressure at the stagnation
point is of special significance
and is given its own name—
stagnation pressure. In
incompressible flows, the
stagnation pressure at a
stagnation point is equal to
the total pressure throughout
the flow field.
Terminology in compressible fluid dynamics[edit]
In a compressible fluid, such
as air, the temperature and
density are essential when
determining the state of the
fluid. In addition to the
concept of total pressure (also
known as stagnation
pressure), the concepts of
total (or stagnation)
temperature and total (or
stagnation) density are also
essential in any study of
compressible fluid flows. To
avoid potential ambiguity
when referring to temperature
and density, many authors
use the terms static
temperature and static
density. Static temperature is
identical to temperature; and
static density is identical to
density; and both can be
identified for every point in a
fluid flow field.
The temperature and density
at a stagnation point are
called stagnation temperature
and stagnation density.
A similar approach is also
taken with the thermodynamic
properties of compressible
fluids. Many authors use the
terms total (or
stagnation) enthalpy and total
(or stagnation) entropy. The
terms static enthalpy and
static entropy appear to be
less common, but where they
are used they mean nothing
more than enthalpy and
entropy respectively, and the
prefix "static" is being used to
avoid ambiguity with their
'total' or 'stagnation'
counterparts. Because the
'total' flow conditions are
defined
by isentropically bringing the
fluid to rest, the total (or
stagnation) entropy is by
definition always equal to the
"static" entropy.
See also[edit]
Fields of study[edit]
Acoustic theory
Aerodynamics
Aeroelasticity
Aeronautics
Computational fluid
dynamics
Flow measurement
Geophysical fluid
dynamics
Hemodynamics
Hydraulics
Hydrology
Hydrostatics
Electrohydrodynamics
Magnetohydrodynamics
Metafluid dynamics
Rheology
Quantum hydrodynamics
Mathematical equations and concepts[edit]
Airy wave theory
Bernoulli's equation
Reynolds transport
theorem
Benjamin–Bona–Mahony
equation
Boussinesq
approximation
(buoyancy)
Boussinesq
approximation (water
waves)
Conservation laws
Euler equations (fluid
dynamics)
Different Types of
Boundary Conditions in
Fluid Dynamics
Darcy's law
Dynamic pressure
Fluid statics
Hagen–Poiseuille
equation
Helmholtz's theorems
Kirchhoff equations
Knudsen equation
Manning equation
Mild-slope equation
Morison equation
Navier–Stokes equations
Oseen flow
Pascal's law
Poiseuille's law
Potential flow
Pressure
Static pressure
Pressure head
Relativistic Euler
equations
Reynolds decomposition
Stokes flow
Stokes stream function
Stream function
Streamlines, streaklines
and pathlines
Types of fluid flow[edit]
Cavitation
Compressible flow
Couette flow
Free molecular flow
Incompressible flow
Inviscid flow
Isothermal flow
Laminar flow
Open channel flow
Secondary flow
Stream thrust averaging
Superfluidity
Supersonic
Transient flow
Transonic
Turbulent flow
Two-phase flow
Fluid properties[edit]
Density
List of hydrodynamic
instabilities
Newtonian fluid
Non-Newtonian fluid
Surface tension
Viscosity
Vapour pressure
Compressibility
Fluid phenomena[edit]
Boundary layer
Coanda effect
Convection cell
Convergence/Bifurcation
Darwin drift
Drag (force)
Hydrodynamic stability
Kaye effect
Lift (force)
Magnus effect
Ocean surface waves
Rossby wave
Shock wave
Soliton
Stokes drift
Turbulence
Thread breakup
Venturi effect
Vortex
Vorticity
Water hammer
Wave drag
Applications[edit]
Acoustics
Aerodynamics
Cryosphere science
Fluid power
Hydraulic machinery
Meteorology
Naval architecture
Oceanography
Plasma physics
Pneumatics
3D computer graphics
Fluid dynamics journals[edit]
Annual Reviews in Fluid
Mechanics
Journal of Fluid
Mechanics
Physics of Fluids
Experiments in Fluids
European Journal of
Mechanics B: Fluids
Theoretical and
Computational Fluid
Dynamics
Computers and Fluids
International Journal for
Numerical Methods in
Fluids
Flow, Turbulence and
Combustion
Miscellaneous[edit]
Important publications in
fluid dynamics
Isosurface
Keulegan–Carpenter
number
Rotating tank
Sound barrier
Beta plane
Immersed boundary
method
Bridge scour
Finite volume method for
unsteady flow
Notes[edit]
1. Jump up^ Eckert,
Michael (2006). The
Dawn of Fluid
Dynamics: A
Discipline Between
Science and
Technology. Wiley.
p. ix. ISBN 3-527-
40513-5.
2. Jump up^ Anderson,
J.D., Fundamentals of
Aerodynamics, 4th
Ed., McGraw–Hill,
2007.
3. Jump up^ White,
F.M., Viscous Fluid
Flow, McGraw–Hill,
1974.
4. Jump up^ Shengtai
Li, Hui Li "Parallel
AMR Code for
Compressible MHD or
HD Equations" (Los
Alamos National
Laboratory) [1]
5. Jump up^ Transient
state or unsteady
state?
6. Jump up^ See Pope
(2000), page 75.
7. Jump up^ See, for
example, Schlatter et
al, Phys. Fluids 21,
051702
(2009); do
i:10.1063/1.3139294
8. Jump up^ See Pope
(2000), page 344.
9. Jump up^ Clancy,
L.J. Aerodynamics,
page 21
References[edit]
Acheson, D. J.
(1990). Elementary Fluid
Dynamics. Clarendon
Press. ISBN 0-19-
859679-0.
Batchelor, G.
K. (1967). An
Introduction to Fluid
Dynamics. Cambridge
University Press. ISBN 0-
521-66396-2.
Chanson,
H. (2009). Applied
Hydrodynamics: An
Introduction to Ideal and
Real Fluid Flows. CRC
Press, Taylor & Francis
Group, Leiden, The
Netherlands, 478
pages. ISBN 978-0-415-
49271-3.
Clancy, L. J.
(1975). Aerodynamics.
London: Pitman
Publishing
Limited. ISBN 0-273-
01120-0.
Lamb,
Horace (1994). Hydrodyn
amics (6th ed.).
Cambridge University
Press. ISBN 0-521-
45868-4. Originally
published in 1879, the
6th extended edition
appeared first in 1932.
Landau, L. D. ; Lifshitz, E.
M. (1987). Fluid
Mechanics. Course of
Theoretical Physics (2nd
ed.). Pergamon
Press. ISBN 0-7506-
2767-0.
Milne-Thompson, L. M.
(1968). Theoretical
Hydrodynamics (5th ed.).
Macmillan. Originally
published in 1938.
Pope, Stephen B.
(2000). Turbulent Flows.
Cambridge University
Press. ISBN 0-521-
59886-9.
Shinbrot, M.
(1973). Lectures on Fluid
Mechanics. Gordon and
Breach. ISBN 0-677-
01710-3.
External links[edit]
Wikimedia Commons has
media related to Fluid
dynamics.
Wikimedia Commons has
media related to Fluid
mechanics.
eFluids , containing
several galleries of fluid
motion
National Committee for
Fluid Mechanics Films
(NCFMF), containing
films on several subjects
in fluid dynamics
(in RealMedia format)
List of Fluid Dynamics
books
V
T
E
V
T
E
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